cup of all-purpose flour, plus more for rolling the dough
½ cup unsweetened Dutch-process cocoa powder
½ tsp baking soda
½ tsp baking powder
½ tsp salt
5 Tbl unsalted butter, room temperature
¾ cup sugar
1 large egg
1 large egg yolk
¾ tsp pure peppermint extract
8 candy canes, or 30 round peppermint candies, crushed
12 oz white chocolate, chips or coarsely chopped
In a medium-sized mixing bowl, whisk together the flour, cocoa powder, baking soda, baking powder and salt. Set aside.
In the bowl of an electric stand mixer, combine the butter and sugar. (You can also use a hand-held electric mixer too.) Beat on medium-high speed until light and fluffy, about 2 minutes. Scrape down the sides of the bowl and the paddle with a rubber spatula, as needed. Reduce the speed to medium-low. Add the egg, then the yolk, beating well after each addition. Beat in the peppermint extract. Slowly add the flour mixture, and beat only until just incorporated. Shape the dough into two equal disks. Wrap in plastic, and refrigerate until firm, at least 1 hour and up to 2 days.
Preheat the oven to 325°F and arrange the racks in the upper and lower third positions. Line 2 baking sheets with parchment paper and set aside.
On a lightly floured surface, roll out one disk of dough to 1/8-inch thickness. Using a 2-inch round cookie cutter, cut out circles, and arrange 1-inch apart on the parchment-lined baking sheets. (Roll and cut the left-over scraps once, then discard the remaining scraps.) Freeze the cookies on their baking sheets until firm, about 15 minutes. Repeat with the remaining disk.
Bake, 2 sheets at a time, until the cookies are dry to the touch, about 12 minutes, rotating the pans (trading their positions and turning them each 180 degrees) about half-way thorough the cooking time. Transfer the parchment with the cookies to wire racks to cool. Continue baking the remaining cookies. Undecorated cookies can be kept at room temperature in an air-tight container for up to 3 days.
Sift the crushed peppermint candy, separating the large pieces from the dust, and reserving both.
Place the white chocolate in a heatproof bowl set over a pot filled with about an inch of gently simmering water. Stir frequently as the chocolate melts. Remove from the heat. Using a small off-set spatula, frost each cookie top with a smooth circle of melted chocolate. Immediately sprinkle with about ¼ teaspoon of either the candy pieces or the dust. Repeat with the remaining cookies, sprinkling half with candy pieces and half with candy dust. Refrigerate on the baking sheets until set, up to 3 hours. Keep refrigerated in an air-tight container. The cookies will keep up to 5 or so days, but are best eaten earlier rather than later.